A simple nerf to last whisper won't solve any issues because the armor values in game are tuned with the expectation that last whisper will be used. Making it purely a % of bonus based model will shrink the effective health difference between tanks and squishies, which after the fallout will simply amount to squishies being a tad tankier, something that the likes of marksmen currently aren't balanced around. You can see how much of an influence barrier and heal have on how effective marksmen are-- now increase their durability by about 25% and you'd get an idea of how swapping last whisper to purely bonus pen would affect the game.
I know you are claiming that the game doesn't really need any flat pen and you'd probably like to remove it at some point, but I disagree.
We need 2 types of penetrations. We need something I can buy at the store-- a multiplier-- that will be most effective against squishy targets, and we need a separate one that will be most effective versus tanky targets.
The idea that flat damage provides a greater increase versus squishy targets than tanky targets is great but it falters mathematically, because the difference isn't great enough even if a % bonus model were applied, because the innate armor values of champions has crawled up and up and now its too high. We had a global armor boost with the removal of dodge, and now we have +4 more in patch 4.5 AND we have tons more options for armor than we did a long time ago. Armor is just too commonplace, and that may be due to the fact that physical damage is just too high.
If innate armor could be low enough that "flat damage provides a greater increase versus squishy targets than tanky targets" would work, it would also create snowballing issues. In particular in games where you are able to get a lead to build raw damage and abuse that damage BEFORE the enemy can build a sufficient amount of armor on their tanks. There would also be issues about what happens when you happen to be ahead as a squishy and you buy one single armor item and now the enemy damage output onto you is trashy, since there is no flat pen option and you do not have enough bonus armor as of yet to make last whisper appealing.
Breaking up armor penetration into a choice between hitting squishies harder or hitting tanks harder is a good thing because it doesn't overload any statistics, provides one lever for one job, and makes the jobs of future live balancers that much easier.
If it were up to me I'd remove armor pen runes, then consolidate flat armor penetration into one or two key items. I'd put a ramping system or some condition based system on it similar to black cleaver in order to provide the squishies with some gameplay other than "get bursted and can't react". I wouldn't copy black cleaver btw, because by the time you apply the stacks they should be dead anyway, leaving the item feeling lackluster. I'd probably tie it to a positional model, where bad positioning on the enemy team can amplify your flat shred.
And then I'd make last whisper bonus based rather than total based, and I'd change it so that flat pen is applied instead of percent pen if it is greater, otherwise percent is applied-- that way you aren't roped into building both end game.
Now building penetration is still something you want to do at some point in the game, but WHICH penetration you build is a choice. You could build percent pen for melting tanks, you could build flat pen for massacring squishies, but if you build both you will pay an item slot cost for having both and they will NOT amplify eachother-- you are NOT encouraged to sacrifice 2 item slots to what is otherwise quite a boring mechanism that only amplifies damage.
I prefer items that offer gameplay or at the very least an actual choice. Reworking pen this way provides a real actual choice with real tangible costs. It still isn't idea though, because its just as mandatory as boots are for the DPS classes, and that isn't something I really agree should be a thing.